Timeline keyframe not showing when changing to different transform rotation representation
The quaternion, euler and axis angle rotations are stored as three different properties, the rotation_mode is used to determine which property is used for rotation.

Yes you will need to re-create the keyframes if you want to use a different rotation mode. You can choose the quaternion rotation mode, move to a keyframe and then change to euler and add a keyframe to the euler values as they will then match the values of the quaternion, then repeat for each keyframe of the rotation. If you have a lot of animation to do then a python script could be used to automate this.